Introduction
Before 1854 when Boole published An
Investigation of The Laws of Thought, on which are Founded the
Mathematical Theories of Logic and Probabilities (henceforth: Laws of
Thought), the subject of ?logic? in the western world
was entirely restricted to the study of the Aristotelian syllogism, which was
as old as 384-322 BC. In this work, Boole introduces a sort of algebra of
propositions with probability. Boole?s logic forms the basis for present day
Boolean Algebra, which in turn lies at the base of computer science. Because
Boole is essentially inventing a number of new concepts, the discussions
concerning his ideas of logic are both accessible to the non-specialist and
fascinating for the historian or philosopher of mathematics and logic.
George Boole (1815 ? 1864) was born
in Lincolnshire, England. He was born of humble origin as the first son of a
shoemaker, but posthumously, he rose to the heights of the moon, by having a
moon crater named after him in 1967. For financial reasons, he did not receive
the best education, but neverthelesshe learned Latin from a local bookseller
and taught himself French, German, and Greek. At the age of nineteen, Boole had
to support his parents and siblings. He started teaching at a local school, and
later set up and ran a boarding school with the help of his family.
Boole only later turned his
attention to mathematics. He was encouraged in his mathematical pursuits by
Duncan Gregory at Cambridge who recognized his deep understanding and
imaginative approach to mathematics. Boole?s early contributions to mathematics
were to calculus and analysis. Boole was more widely acclaimed for his
contributions to mathematics in 1844, when he received the Royal Medal from the
Royal Society for his paper On a General Method of Analysis. In 1849,
Boole was appointed chair of mathematics at Queens College, Cork. In 1854,
while teaching at Cork, Boole published his most important work The Laws of
Thought:
I am now about to set seriously to work
upon preparing for the press an account of my theory of Logic and Probabilities
which in its present state I look upon as the most valuable if not the only
valuable contribution that I have made or am likely to make to Science?.
Writing to his printers, De Morgan writes that Boole ?is meditating
typographically on his mathematical logic, which is a very original thing, and,
for power of thought, worthy to be printed?.? Boole had an important
correspondence with De Morgan, and his logical innovations have inspired many
logicians and mathematicians since, including Charles Saunders Pierce and J.
Venn, inventor to the famous Venn diagrams.
Boole?s most famous works are ?The
Mathematical Analysis of Logic,? published in 1847 and The Laws of Thought.
It is from these two works that we learn some of Boole?s most important and
influential ideas. One of Boole?s great and original ideas was to claim logic
as an area of mathematics, where, previously, logic had belonged exclusively to
philosophy. This freed up the conception of what logic is and allowed Boole to
then introduce a new structure to logical reasoning. In Laws of Thought,
Boole braids together three ideas: that logic should account for, and capture,
the syllogistic reasoning while reaching beyond it, that there are law-like
constraints on our reasoning, and that reasoning about probabilities is a
logical primitive.
Boole invented, what is now called
?Boolean algebra.? He saw his logic as a formal representation of laws of
reasoning, in the sense of marshalling our reasoning. One of Boole?s greatest
innovations was to think of algebra as not only pertaining to number, but to
other things such as terms or propositions. In his logical system, Boole
introduces a notion of class. From this, Boole very naturally develops the
notions of sub-class, the intersection of two classes, the union of two classes
and the complement of a class. Today, these are the familiar Boolean
connectives: IF THEN, AND, OR, NOT. Boole also introduces notions of
quantification and probability. Semantics are captured by the numbers ?zero?
and ?one? to signify ?nothing? and ?the universe,? respectively. This elegantly
simple concept of semantics has significantly contributed to the shape of
modern mathematics and computer science, where we re-interpret ?zero? and ?one?
to mean ?off / on? or ?false / true.?
It is thanks to Boole?s Laws of
Thought, that the topic of ?logic? was released from the constraints of the
syllogism by applying concepts of proof from algebra to terms and propositions.
This allowed logical proofs to be extended to include more than two premises.
This is important because here we have the first glimpse of a notion of logical
proof in which each step does not have to be obviously related to the
conclusion. Logical proof is newly understood as a procedure that can take
several steps, each one of which follows from pre-accepted principles of
deduction. These principles of proof are taken from algebra and probability
theory, which Boole thought epitomized standards of good human reasoning.
Charles Sanders Pierce (1839?1914)
was impressed by Boole?s contribution to mathematics and worked on the
electrical application of the Boolean logic. This became one of the pre-cursors
of electrical computing machines. Pierce also taught one of the first courses
in Boolean algebra. Pierce developed Boole?s logic further by making explicit
the notion of truth-value assignment to propositions. He added the idea that a
necessary truth is one that is true under any truth-value assignment. With this
we have the stage set for the truth-table definitions of the logical
connectives. These were developed independently by Post and Wittgenstein in the
early 1920s.
Despite his monumental achievements, Boole?s Laws of Thought
is too-little read. One reason for this is that Boole?s logic was greatly
surpassed in 1879 with the publication of Frege?s Begriffsschrift. In
this, Frege develops a logic of predicates, functions, and relations, with
quantifiers quantifying over first and second-order variables. Frege?s system
is clearer and much more supple and sophisticated than Boole?s. Indeed, apart
from the notation, the logic we learn today, in the form of predicate logic or
first-order logic, is descended from Frege. Lying in Frege?s shadow, Boole?s Laws
of Thought has been too often dismissed as being, at best, of historical
interest.
A second reason Boole is overlooked
is that he was heavily criticized by Frege and Russell for being
psychologistic, where this is understood in the standard sense of logic being
essentially a mental construct, and thus, culpable of being subjective. This
criticism is partly based on a confusion of the term. Bornet speculates that,
despite his criticism of the work, Russell never read Boole?s Laws of
Thought! But he did read the title, and it is based on this that Russell
dismisses the work as psychologistic, thinking that it dealt with a description
of how we in fact reason rather than with logic, which tells us how we ought to
reason. Russell?s dismissal was enough to dissuade many potential readers.
Far from being psychologistic, in
the ?mental construct? sense, one can detect hints of logicism and formalism in
The Laws of Thought; where logicism is understood as the reduction of
arithmetic to logic, and formalism is the idea that mathematics is essentially
symbol manipulation. Formalism serves computer science well. By allowing long
chains of reasoning in his logic, and allowing a mechanistic element in the
reasoning, Boole anticipated the computer?s ability to carry out very long proofs.
Boole?s Laws of Thought is
worthy of our careful re-examination. The book is of particular interest to any
serious scholar of the philosophy of, or the history of, logic; anyone
interested in the early history of the computer and notions of automated
computation; anyone interested in the history of probability theory; and anyone
interested in nineteenth-century mathematics. The wide variety of topics
discussed makes Boole?s Laws of Thought well worth the read.
More specifically, philosophers of
logic will be particularly interested in the first, thirteenth, fifteenth, and
final chapter to discover what Boole thought was the importance of logic. In
these chapters we see Boole?s formal system as a formal representation of the
structure of reasoning. He takes logic to be a part of the philosophy of mind,
which in turn is a subspecies of metaphysics. He is also very keen to show the
allegiance of his logic to traditional Aristotelian syllogistic reasoning. In
chapter thirteen he shows how one can recapture the Aristotelian logic in his
formal system. At the time, this was deemed essential to uphold the claim that
what he was discussing was logic, as opposed to mathematics. In chapter
fifteen, Boole goes well beyond the syllogism and uses his formal system to
analyze long arguments by Clarke and Spinoza, both noted for their careful
philosophical arguments. To his credit, Boole is aware of the limitations of
formalizing arguments originally written in a natural language. Not only are
there problems in loyalty of representation of the basic propositions, but also
of the connections made between them. Put another way, Boole is aware that,
showing that one proposition is not derivable from another in the formal system
under a particular translation, using only logical rules of inference, does not
imply that there is no legitimate metaphysical connection between the
propositions. Nevertheless, when philosophical arguments are made as
meticulously as they are in the texts of Clarke and Spinoza, these lend
themselves naturally to further elucidation through logical analysis. Such an
analysis would have been almost impossible if restricted to Aristotelian
syllogistic reasoning.
For those readers interested in the
logical system minus probability, either for historical or conceptual reasons,
you will be interested in chapters two through twelve. In these we have a
charming and candid discussion of some basic logical concepts, some of which
are seldom so described. Boole?s notion of quantity is quirky by today?s
standards. Boole has a general operator, symbolized by the Greek letter ?n.?
This is placed to the left of a term or proposition variable or constant. The
letter ?n?
can be interpreted as ?all? or ?not all,? i.e., ?some.? This has to be
specified independently. For, what interests Boole is the algebra of the
operator, and this remains the same under either interpretation.
Boole discusses probability theory
in chapters sixteen to twenty-one. He not only sums up what has already been
established in probability theory by mathematicians such as Laplace and
Poisson. He brings his own innovative contribution by combining probability
theory with propositional logic. In doing this, Boole is able to discuss the
probability of the truth of a proposition, as opposed to the more usual
?probability of an event.? The merging of these two notions of proposition and
probability is what allows Boole to combine various events, with attending
probabilities, into one (compound) event with one probability. Formerly, these
each had to be treated in isolation. Chapters nineteen to twenty-one give a
more philosophical and general discussion of probability theory in the contexts
of statistics, causation, and judgment. Boole is very careful to distinguish
subjective probability from objective probability, although he does not use
this convenient modern vocabulary.
As we can see, Boole?s Laws of
Thought is a monumental contribution to human thinking. Not only does Boole
venture into several areas which we would normally keep separate today, but,
for him, these areas merge and form a coherent whole, each part informing
another, and all this through deep philosophical reflection. Moreover, the deep
philosophical reflection is not a metaphysical treatise made in isolation of
any ?hard data.? Boole?s technical contribution is both original and
substantial.
Michele Friend is Assistant Professor at The George
Washington University. She teaches courses in formal logic, critical thinking,
and the philosophy of mathematics. Her research is in the philosophy of
mathematics and logic.